1. Technical Field
The invention relates to predicting the outcome of a chaotic system. More particularly, the invention relates to predicting outcome of chaotic systems using Lyapunov exponents.
2. Description of the Related Art
Chaotic non-linear systems are known. Examples of chaotic systems include, but are not limited to, the weather, the stock market and fluid flow. By its nature, it is difficult to calculate the outcome of a chaotic system. A slight change in initial conditions or an intervening event can result in a completely different outcome or trajectory. In a weather system, a slight change in the initial conditions of temperatures and pressures at various locations results in a completely different weather trajectory. As a result, a long range forecast may be difficult or impossible to make. One way that chaotic systems can be defined is by calculating a Lyapunov exponent, which is a measure of the rate of divergence of a trajectory with time.
It is well known that the significant digits of variables defining a chaotic system are rapidly lost during calculation and the initially valid digits in the chaotic system become meaningless in a short number of iterations. The number of valid digits n(t) decreases linearly with time. The maximal Lyapunov exponent L is equal to—dn/dt and can be calculated to determine the number of digits which are lost per unit time. Traditional Lyapunov exponent algorithms, such as those calculated by the Benettin technique, can be applied only to a system of differential equations. A different algorithm is used for discrete maps and until now no algorithm existed for any other mathematical process.